| L(s) = 1 | + (0.342 − 0.939i)2-s + (−1.41 + 0.994i)3-s + (−0.766 − 0.642i)4-s + (−0.622 + 3.53i)5-s + (0.449 + 1.67i)6-s + (−0.609 − 2.57i)7-s + (−0.866 + 0.500i)8-s + (1.02 − 2.82i)9-s + (3.10 + 1.79i)10-s + (−4.15 + 0.732i)11-s + (1.72 + 0.149i)12-s + (−2.25 − 6.20i)13-s + (−2.62 − 0.307i)14-s + (−2.62 − 5.62i)15-s + (0.173 + 0.984i)16-s + (2.59 − 4.49i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.818 + 0.574i)3-s + (−0.383 − 0.321i)4-s + (−0.278 + 1.57i)5-s + (0.183 + 0.682i)6-s + (−0.230 − 0.973i)7-s + (−0.306 + 0.176i)8-s + (0.340 − 0.940i)9-s + (0.982 + 0.566i)10-s + (−1.25 + 0.220i)11-s + (0.498 + 0.0432i)12-s + (−0.626 − 1.72i)13-s + (−0.702 − 0.0821i)14-s + (−0.678 − 1.45i)15-s + (0.0434 + 0.246i)16-s + (0.630 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0191466 - 0.185558i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0191466 - 0.185558i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 + 0.939i)T \) |
| 3 | \( 1 + (1.41 - 0.994i)T \) |
| 7 | \( 1 + (0.609 + 2.57i)T \) |
| good | 5 | \( 1 + (0.622 - 3.53i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (4.15 - 0.732i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.25 + 6.20i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.59 + 4.49i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.98 - 2.30i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.69 - 3.21i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 4.35i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.80 - 2.15i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.76 - 4.78i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.08 - 1.12i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.856 + 4.85i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (0.00676 - 0.00567i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 3.95iT - 53T^{2} \) |
| 59 | \( 1 + (0.405 - 2.29i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.23 - 1.46i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.97 + 0.719i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (2.82 + 1.63i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.15 + 4.70i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.5 + 3.83i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.26 + 0.459i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (5.76 + 9.98i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.42 - 0.427i)T + (91.1 - 33.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.70402549737990185761766969852, −10.22426015346976529596150744622, −9.971843982009218038119208144525, −7.87315945119326137043101494163, −7.13929803648362693099121476379, −5.92787721720513291558948925030, −4.93244780545598763609203459413, −3.60812822928181619189705842206, −2.83562059114614780525197473831, −0.12052994733160876068357818629,
2.01983512965910182617342299654, 4.35138580772690575998209230214, 5.12994514568352256093903833286, 5.89039637312514533152791841893, 6.94030116176206865347273197125, 8.194433243067498954575028586439, 8.644672760415436300668343885779, 9.787822129476569182389534422531, 11.15882657980944246965303512844, 12.24237209787668544512791949818
